The New Macroeconometrics:

A Bayesian Approach�

Jesús Fernández-Villaverde

University of Pennsylvania, NBER, and CEPR

Pablo Guerrón-Quintana

North Carolina State University

Juan F. Rubio-Ramírez

Duke University and Federal Reserve Bank of Atlanta

January 19, 2009

�Corresponding author: Juan F. Rubio-Ramírez, 213 Social Sciences, Duke University, Durham, NC 27708,USA. E-mail: Juan.Rubio-Ramirez@duke.edu. We thank Alex Groves for his help with preparing the dataused in the estimation. Beyond the usual disclaimer, we must note that any views expressed herein arethose of the authors and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal ReserveSystem. Finally, we also thank the NSF for �nancial support.

1

Abstract

This chapter studies the dynamics of the U.S. economy over the last �fty years via

the Bayesian analysis of dynamic stochastic general equilibrium (DSGE) models. Our

application is of particular interest because modern macroeconomics is centered around

the construction of DSGE models. We formulate and estimate a benchmark DSGE

model that captures well the dynamics of the data. This model can be easily applied to

policy analysis at public institutions, such as central banks, private organizations, and

businesses. We explain how to solve the model and how to evaluate the likelihood using

�ltering theory. We also discuss the role of priors and how pre-sample information is

key for a successful understanding of macro data. Our results document a fair amount

of real and nominal rigidities in the U.S. economy. We �nish by pointing out lines for

future research.

2

1. Introduction

This chapter studies the dynamics of the U.S. economy over the last �fty years via the

Bayesian analysis of dynamic stochastic general equilibrium (DSGE) models. Our data are

aggregate, quarterly economic variables and our approach combines macroeconomics (the

study of aggregate economic variables like output or in�ation) with econometrics (the appli-

cation of formal statistical tools to economics). This is an example application in what is

often called the New Macroeconometrics.

Our application is of particular interest because modern macroeconomics is centered

around the construction of DSGE models. Economists rely on DSGE models to organize

and clarify their thinking, to measure the importance of economic phenomena, and to pro-

vide policy prescriptions. DSGE models start by specifying a number of economic agents,

typically households, �rms, a government, and often a foreign sector, and embodying them

with behavioral assumptions, commonly the maximization of an objective function like utility

or pro�ts.

The inelegant name of DSGE comes from what happens next. First, economists postulate

some sources of shocks to the model: shocks to productivity, to preferences, to taxes, to mon-

etary policy, etc. Those shocks are the �stochastic�part that will drive the model. Second,

economists study how agents make their decisions over time as a response to these shocks

(hence �dynamic�). Finally, economists focus on the investigation of aggregate outcomes,

thus �general equilibrium.�

Before we discuss our application, it is important to emphasize that economists employ the

word �equilibrium�in a particular and precise technical sense, which is di¤erent from daily use

or from its meaning in other sciences. We call equilibrium a situation where a) the agents in

the model follow a concrete behavioral assumption (usually, but not necessarily, maximization

of either their utility or of pro�ts) and b) the decisions of the agents are consistent with each

other (for example, the number of units of the good sold must be equal to the number of units

of the good bought). Nothing in the de�nition of equilibrium implies that prices or allocations

of goods are constant over time (they may be increasing, decreasing, or �uctuating in rather

arbitrary ways), or that the economy is at a rest point, the sense in which equilibrium is

commonly used in other �elds, particularly in the natural sciences. Many misunderstandings

about the accomplishments and potentialities of the concept of equilibrium in economics come

about because of a failure to appreciate the subtle di¤erence in language across �elds.

General equilibrium has a long tradition in economics, being already implicit in the work

of the French Physiocrats of the 18th century, such as Cantillon, Turgot, or Quesnay, or in

Adam Smith�s magnum opus, The Wealth of Nations. However, general equilibrium did not

3

take a front seat in economics until 1874 when Léon Walras published a pioneering book,

Elements of Pure Economics, in which he laid down the foundations of modern equilibrium

theory.

For many decades after Walras, general equilibrium theory focused on static models. This

was not out of a lack of appreciation by economists of the importance of dynamics, which were

always at the core of practical concerns, but because of the absence of proper methodological

instruments to investigate economic change.

Fortunately, the development of recursive techniques in applied mathematics (dynamic

programming, Kalman �ltering, and optimal control theory, among others) during the 1950s

and 1960s provided the tools that economists required for the formal analysis of dynamics.

These new methods were �rst applied to the search for microfoundations of macroeconomics

during the 1960s, which attempted to move from ad hoc descriptions of aggregate behav-

ior motivated by empirical regularities to descriptions based on �rst principles of individual

decision-making. Later, in the 1970s, recursive techniques opened the door to the rational

expectations revolution, started by John Muth (1961) and Robert Lucas (1972), who high-

lighted the importance of specifying the expectations of the agents in the model in a way

that was consistent with their behavior.

This research culminated in 1982 with an immensely in�uential paper by Finn Kydland

and Edward Prescott, Time to Build and Aggregate Fluctuations.1 This article presented

the �rst modern DSGE model, on this propitious occasion, one tailored to explaining the

�uctuations of the U.S. economy. Even if most of the material in the paper was already

present in other articles written in the previous years by leading economists like Robert Lucas,

Thomas Sargent, Christopher Sims, or Neil Wallace, the genius of Kydland and Prescott

was to mix the existing ingredients in such a path-breaking recipe that they redirected the

attention of the profession to DSGE modelling.

Kydland and Prescott also opened several methodological discussions about how to best

construct and evaluate dynamic economic models. The conversation that concerns us in this

chapter was about how to take the models to the data, how to assess their behavior, and,

potentially, how to compare competing theories. Kydland and Prescott were skeptical about

the abilities of formal statistics to provide a useful framework for these three tasks. Instead,

they proposed to �calibrate�DSGE models, i.e., to select parameter values by matching some

moments of the data and by borrowing from microeconomic evidence, and to judge models by

their ability to reproduce properties of the data that had not been employed for calibration

(for example, calibration often exploits the �rst moments of the data to determine parameter

1The impact of this work was recognized in 2004 when Kydland and Prescott received the Nobel Prize ineconomics for this paper and some previous work on time inconsistency in 1977.

4

values, while the evaluation of the model is done by looking at second moments).

In our previous paragraph, we introduced the idea of �parameters of the model.�Before

we proceed, it is worthwhile to discuss further what these parameters are. In any DSGE

economy, there are functions, like the utility function, the production function, etc. that

describe the preferences, technologies, and information sets of the agents. These functions

are indexed by a set of parameters. One of the attractive features of DSGE models in the eyes

of many economists is that these parameters carry a clear behavioral interpretation: they are

directly linked to a feature of the environment about which we can tell an economic history.

For instance, in the utility function, we have a discount factor, which tells us how patient

the households are, and a risk aversion, which tells us how much households dislike uncer-

tainty. These behavioral parameters (also called �deep parameters�or, more ambiguously,

�structural parameters�) are of interest because they are invariant to interventions, including

shocks by nature or, more important, changes in economic policy.2

Kydland and Prescott�s calibration became popular because, back in the early 1980s,

researchers could not estimate the behavioral parameters of the models in an e¢ cient way.

The bottleneck was how to evaluate the likelihood of the model. The equilibrium dynamics of

DSGE economies cannot be computed analytically. Instead, we need to resort to numerical

approximations. One key issue is, then, how to go from this numerical solution to the

likelihood of the model. Nowadays, most researchers apply �ltering theory to accomplish

this goal. Three decades ago, economists were less familiar with �lters and faced the speed

constraints of existing computers. Only limited estimation exercises were possible and even

those were performed at a considerable cost. Furthermore, as recalled by Sargent (2005), the

early experiments on estimation suggested that the prototype DSGE models from the early

1980s were so far away from �tting the data that applying statistical methods to them was

not worth the time and e¤ort. Consequently, for over a decade, little work was done on the

estimation of DSGE models.3

2Structural parameters stand in opposition to reduced-form parameters, which are those that index em-pirical models estimated with a weaker link with economic theory. Many economists distrust the evaluationof economic policy undertaken with reduced parameters because they suspect that the historical relationsbetween variables will not be invariant to the changes in economic policy that we are investigating. Thisinsight is known as the Lucas�critique, after Robert Lucas, who forcefully emphasized this point in a renown1976 article (Lucas, 1976).

3There was one partial exception. Lars Peter Hansen proposed the use of the Generalized Method ofMoments, or GMM (Hansen, 1982), to estimate the parameters of the model by minimizing a quadraticform built from the di¤erence between moments implied by the model and moments in the data. The GMMspread quickly, especially in �nance, because of the simplicity of its implementation and because it was thegeneralization of well-known techniques in econometrics such as instrumental variables OLS. However, theGMM does not deliver all the powerful results implied by likelihood methods and it often has a disappointingperformance in small samples.

5

Again, advances in mathematical tools and in economic theory rapidly changed the land-

scape. From the perspective of macroeconomics, the streamlined DSGE models of the 1980s

begot much richer models in the 1990s. One remarkably successful extension was the intro-

duction of nominal and real rigidities, i.e., the conception that agents cannot immediately

adjust to changes in the economic environment. In particular, many of the new DSGE models

focused on studying the consequences of limitations in how frequently or how easily agents

can changes prices and wages (prices and wages are �sticky�). Since the spirit of these models

seemed to capture the tradition of Keynesian economics, they quickly became known as New

Keynesian DSGE models (Woodford, 2003). By capturing these nominal and real rigidities,

New Keynesian DSGE models began to have a �ghting chance at explaining the dynamics

of the data, hence suggesting that formally estimating them could be of interest. But this

desire for formal estimation would have led to naught if better statistical/numerical methods

had not become widely available.

First, economists found out how to approximate the equilibrium dynamics of DSGE mod-

els much faster and more accurately. Since estimating DSGEmodels typically involves solving

for the equilibrium dynamics thousands of times, each for a di¤erent combination of para-

meter values, this advance was crucial. Second, economists learned how to run the Kalman

�lter to evaluate the likelihood of the model implied by the linearized equilibrium dynamics.

More recently, economists have also learned how to apply sequential Monte Carlo (SMC)

methods to evaluate the likelihood of DSGE models where equilibrium dynamics is nonlinear

and/or the shocks in the model are not Gaussian. Third, the popularization of Markov chain

Monte Carlo (McMc) algorithms has facilitated the task of exploring and characterizing the

likelihood of DSGE models.

Those advances have resulted in an explosion of the estimation of DSGE models, both at

the academic level (An and Schorfheide, 2006) and, more remarkable still, at the institutional

level, where a growing number of policy-making institutions are estimating DSGE models for

policy analysis and forecasting (see appendix A for a partial list). Furthermore, there is

growing evidence of the good forecasting record of DSGE models, even if we compare them

with the judgmental predictions that sta¤ economists prepare within these policy-making

institutions relying on real-time data (Edge, Kiley, and Laforte, 2008).

One of the features of the New Macroeconometrics that the readers of this volume will �nd

exciting is that the (overwhelming?) majority of it is done from an explicitly Bayesian per-

spective. There are several reasons for this. First, priors are a natural device for economists.

To begin with, they use them extensively in game theory or in learning models. More to the

point, many economists feel that they have reasonably concrete beliefs about plausible values

for most of the behavioral parameters, beliefs built perhaps from years of experience with

6

models and their properties, perhaps from introspection (how risk adverse am I?), perhaps

from well-educated economic intuition. Priors o¤er researchers a �exible way to introduce

these beliefs as pre-sample information.

Second, DSGE models are indexed by a relatively large number of parameters (from

around 10 to around 60 or 70 depending on the size of the model) while data are sparse (in

the best case scenario, the U.S. economy, we have 216 quarters of data from 1954 to 2007).4

Under this low ratio data/parameters, the desirable small sample properties of Bayesian

methods and the inclusion of pre-sample information are peculiarly attractive.

Third, and possibly as a consequence of the argument above, the likelihoods of DSGE

models tend to be wild, with dozens of maxima and minima that defeat the best optimization

algorithms. McMc are a robust and simple procedure to get around, or at least alleviate,

these problems (so much so, that some econometricians who still prefer a classical approach to

inference recommend the application of McMc algorithms by adopting a �pseudo-bayesian�

stand; see Chernozhukov and Hong, 2003). Our own experience has been that models that we

could estimate in a few days using McMc without an unusual e¤ort turned out to be extremely

di¢ cult to estimate using maximum likelihood. Even the most pragmatic of economists,

with the lowest possible interest in axiomatic foundations of inference, �nds this feasibility

argument rather compelling.

Fourth, the Bayesian approach deals in a transparent way with misspeci�cation and iden-

ti�cation problems, which are pervasive in the estimation of DSGE models (Canova and Sala,

2006). After all, a DSGE model is a very stylized and simpli�ed view of the economy that

focuses only on the most important mechanisms at play. Hence, the model is false by con-

struction and we need to keep this notion constantly in view, something that the Bayesian

way of thinking has an easier time with than frequentist approaches.

Finally, Bayesian analysis has a direct link with decision theory. The connection is par-

ticularly relevant for DSGE models since they are used for policy analysis. Many of the

relevant policy decisions require an explicit consideration of uncertainty and asymmetric risk

assessments. Most economists, for example, read the empirical evidence as suggesting that

the costs of a 1% de�ation are considerably bigger than the costs of 1% in�ation. Hence,

a central bank with a price stability goal (such as the Federal Reserve System in the U.S.

or the European Central Bank) faces a radically asymmetric loss function with respect to

deviations from the in�ation target.

4Data before 1954 are less reliable and the structure of the economy back then was su¢ ciently di¤erentas to render the estimation of the same model problematic. Elsewhere (Fernández-Villaverde and Rubio-Ramírez, 2008), we have argued that a similar argument holds for data even before 1981. For most countriesoutside the U.S., the situation is much worse, since we do not have reliable quarterly data until the 1970s oreven the 1980s.

7

To illustrate this lengthy discussion, we will present a benchmark New Keynesian DSGE

model, we will estimate it with U.S. data, and we will discuss our results. We will close by

outlining three active areas of research in the estimation of DSGE models that highlight some

of the challenges that we see ahead of us for the next few years.

2. A Benchmark New Keynesian Model

Due to space constraints, we cannot present a benchmark New Keynesian DSGE model in all

its details. Instead, we will just outline its main features to provide a �avor of what the model

is about and what it can and cannot deliver. In particular, we will omit many discussions of

why are we doing things the way we do. We will ask the reader to trust that many of our

choices are not arbitrary but are the outcome of many years of discussion in the literature. The

interested reader can access the whole description of the model at a complementary technical

appendix posted at www.econ.upenn.edu/~jesusfv/benchmark_DSGE.pdf. The model we

select embodies what is considered the current standard of New Keynesian macroeconomics

(see Christiano, Eichenbaum, and Evans, 2005) and it is extremely close to the models being

employed by several central banks as inputs for policy decisions.

The agents in the model will include a) households that consume, save, and supply labor

to a labor �packer,�b) a labor �packer� that puts together the labor supplied by di¤erent

households into an homogeneous labor unit, c) intermediate good producers, who produce

goods using capital and aggregated labor, d) a �nal good producer that mixes all the inter-

mediate goods into a �nal good that households consume or invest in, and e) a government

that sets monetary policy through open market operations. We present in turn each type of

agents.

2.1. Households

There is a continuum of in�nitely lived households in the economy indexed by j. The house-

holds maximize their utility function, which is separable in consumption, cjt, real money

balances (nominal money, mojt; divided by the aggregate price level; pt); and hours worked,

ljt:

E01Xt=0

�tedt

(log (cjt � hcjt�1) + � log

�mojtpt

�� e't

l1+#jt

1 + #

)where � is the discount factor, h controls habit persistence, # is the inverse of the Frisch labor

supply elasticity (how much labor supply changes when the wage changes while keeping

consumption constant), dt is a shock to intertemporal preference that follows the AR(1)

8

process:

dt = �ddt�1 + e�d"d;t where "d;t � N (0; 1);

and 't is a labor supply shock that also follows an AR(1) process:

't = �''t�1 + e�'"';t where "';t � N (0; 1):

These two shocks, "d;t and "';t, are equal across all agents.

Households trade on assets contingent on idiosyncratic and aggregate events. By ajt+1 we

indicate the amount of those securities that pay one unit of consumption at time t+1 in event

!j;t+1;t purchased by household j at time t at real price qjt+1;t. These one-period securities are

su¢ cient to span the whole set of �nancial contracts regardless of their duration. Households

also hold an amount bjt of government bonds that pay a nominal gross interest rate of Rtand invest a quantity xt of the �nal good. Thus, the j � th household�s budget constraint is

given by:

cjt + xjt +mojtpt

+bjt+1pt

+

Zqjt+1;tajt+1d!j;t+1;t

= wjtljt +�rtujt � ��1t � [ujt]

�kjt�1 +

mojt�1pt

+Rt�1bjtpt+ ajt + Tt +zt (1)

where wjt is the real wage, rt the real rental price of capital, ujt > 0 the intensity of use of

capital, ��1t � [ujt] is the physical cost of ujt in resource terms (where � [u] = �1 (u� 1) +�22(u � 1)2 and �1;�2 � 0), �t is an investment-speci�c technological shock that we will

describe momentarily, Tt is a lump-sum transfer from the government, andzt is the householdshare of the pro�ts of the �rms in the economy.

Given a depreciation rate � and a quadratic adjustment cost function

V

�xtxt�1

�=�

2

�xtxt�1

� �x�2

;

where � � 0 and �x is the long-run growth of investment, capital evolves as:

kjt = (1� �) kjt�1 + �t

�1� V

�xjtxjt�1

��xjt:

Note that in this equation there appears an investment-speci�c technological shock �t that

also follows an autoregressive process:

�t = �t�1 exp (�� + z�;t) where z�;t = ��"�;t and "�;t � N (0; 1):

9

Thus, the problem of the household is a Lagrangian function formed by the utility function,

the law of motion for capital, and the budget constraint. The �rst order conditions of this

problem with respect to cjt, bjt, ujt, kjt, and xjt are standard (although messy), and we refer

the reader to the online appendix for a detailed exposition.

The �rst order condition with respect to labor and wages is more involved because, as we

explained in the introduction, households will face rigidities in changing their wages.5 Each

household supplies a slightly di¤erent type of labor service (for example, some households

write chapters on Bayesian estimation of DSGE models and some write chapters on hierar-

chical models) that is aggregated by a labor �packer� (in our example, an editor) into an

homogeneous labor unit (�Bayesian research�) according to the function:

ldt =

�Z 1

0

l��1�

jt dj

� ���1

(2)

where � controls the elasticity of substitution among di¤erent types of labor and ldt is the

aggregate labor demand. The technical role that the labor �packer�plays in the model is

to allow for the existence of many di¤erent types of labor, and hence for the possibility of

di¤erent wages, while keeping the heterogeneity of agents at a tractable level.

The labor �packer� takes all wages wjt of the di¤erentiated labor and the wage of the

homogeneous labor unit wt as given and maximizes pro�ts subject to the production function

(2). Thus, the �rst order conditions of the labor �packer�are:

ljt =

�wjtwt

���ldt 8j (3)

We assume that there is free entry into the labor packing business. After a few substitutions,

we get an expression for the wage wt as a function of the di¤erent wages wjt:

wt =

�Z 1

0

w1��jt dj

� 11��

:

The way in which households set their wages is through a device called Calvo pricing (for

the economist Guillermo Calvo, who proposed this formulation in 1983). In each period, the

household has a probability 1� �w of being able to change its wages. Otherwise, it can only

5In this type of model, households set up their wages and �rms decide how much labor to hire. Wecould also have speci�ed a model where wages are posted by �rms and households decide how much to work.For several technical reasons, our modelling choice is easier to handle. If the reader has problems seeing ahousehold setting its wage, she can think of the case of an individual contractor posting a wage for hourworked, or a union negotiating a contract in favor of its members.

10

partially index its wages by a fraction �w 2 [0; 1] of past in�ation. Therefore, if a householdhas not been able to change its wage for � periods, its real wage is

�Ys=1

��wt+s�1�t+s

wjt;

the original wage times the indexation divided by the accumulated in�ation. This probability

1� �w represents a streamlined version of a more subtle explanation of wage rigidities (basedon issues like contracts costs, Caplin and Leahy, 1991, or information limitations, Sims, 2002),

which we do not �esh out entirely in the model to keep tractability.6

The average duration of a wage decision in this economy will be 1= (1� �w) ; although

all wages change period by period because of the presence of indexation (in that sense the

economy would look surprisingly �exible to a naïve observer!). Since a suitable law of large

numbers holds for this economy, the probability of changing wages, 1� �w, will also be equalto the fraction of households reoptimizing their wages in each period.

Calvo pricing implies three results. First, the form of the utility function that we selected

and the presence of complete asset markets deliver the remarkable result that all households

that choose their wage in this period will choose exactly the same wage, w�t (also, the con-

sumption, investment, and Lagrangian multiplier �t of the budget constraint of all households

will be the same and we drop the subindex j when no confusion arises):

Second, in every period, a fraction 1 � �w of households set w�t as their wage, while the

remaining fraction �w partially index their price by past in�ation. Consequently, the real

wage index evolves as:

w1��t = �w

���wt�1�t

�1��w1��t�1 + (1� �w)w

�1��t :

Third, after a fair amount of algebra, we can show that the evolution of w�t and an auxiliary

variable ft are governed by two recursive equations:

ft =� � 1�

(w�t )1�� �tw

�t ldt + ��wEt

���wt

�t+1

�1�� �w�t+1w�t

���1ft+1

and:

ft = dt't

�wtw�t

��(1+#) �ldt�1+#

+ ��wEt���wt

�t+1

���(1+#)�w�t+1w�t

��(1+#)ft+1:

6There is, however, a discussion in the literature regarding the potential shortcomings of Calvo pricing.See, for example, Dotsey, King, and Wolman (1999), for a paper that prefers to be explicit about the problemfaced by agents when changing prices.

11

2.2. The Final Good Producer

The �nal good producer aggregates all intermediate goods into a �nal good with the following

production function:

ydt =

�Z 1

0

y"�1"

it di

� ""�1

: (4)

where " controls the elasticity of substitution.

The role of the �nal good producer is to allow for the existence of many �rms with

di¤erentiated products while keeping a simple structure in the heterogeneity of products and

prices. Also, the �nal good producer takes as given all intermediate goods prices pti and the

�nal good price pt; which implies a demand function for each good

yit =

�pitpt

��"ydt 8i;

where ydt is the aggregate demand and by a zero pro�t condition:

pt =

�Z 1

0

p1�"it di

� 11�"

:

2.3. Intermediate Good Producers

There is a continuum of intermediate good producers. Each intermediate good producer i has

access to a technology described by a production function of the form yit = Atk�it�1

�ldit�1���

�zt where kit�1 is the capital rented by the �rm, ldit is the amount of the homogeneous labor

input rented by the �rm, the parameter � corresponds to the �xed cost of production, and

where At follows a unit root process in logs, At = At�1 exp (�A + zA;t) where zA;t = �A"A;t

and "A;t � N (0; 1):The �xed cost � is scaled by the variable zt = A

11��t �

�1��t to guarantee that economic

pro�ts are roughly equal to zero. The variable zt evolves over time as zt = zt�1 exp (�z + zz;t)

where zz;t =zA;t+�z�;t

1�� and �z =�A+���1�� . We can also prove that �z is the mean growth rate

of the economy.

Intermediate good producers solve a two-stage problem. First, given wt and rt, they rent

ldit and kit�1 to minimize real costs of production, which implies a marginal cost of:

mct =

�1

1� �

�1���1

�

��w1��t r�tAt

This marginal cost does not depend on i: all �rms receive the same shocks and rent inputs

at the same price.

12

Second, intermediate good producers choose the price that maximizes discounted real

pro�ts under the same Calvo pricing scheme as households. The only di¤erence is that

now the probability of changing prices is given by 1 � �p and the partial indexation by the

parameter � 2 [0; 1]. We will call p�t the price that intermediate good producers select whenthey are allowed to optimize their prices at time t.

Again, after a fair amount of manipulation, we �nd that the price index evolves as:

p1�"t = �p���t�1

�1�"p1�"t�1 + (1� �p) p

�1�"t

and that p�t is determined by two recursive equations in the auxiliary variable g1t and g

2t :

g1t = �tmctydt + ��pEt

���t�t+1

��"g1t+1

g2t = �t��tydt + ��pEt

���t�t+1

�1�"���t��t+1

�g2t+1

where "g1t = ("� 1)g2t and ��t = p�t=pt.

2.4. The Government

The government plays an extremely limited role in this economy. It sets the nominal interest

rates according to the following policy rule:

RtR=

�Rt�1R

� R0@��t�

� �0@ ydtydt�1

�z

1A y1A1� R

emt (5)

The policy is implemented through open market operations that are �nanced with lump-

sum transfers Tt to ensure that the government budget is balanced period by period. The

variable � is the target level of in�ation, R the steady-state gross return of capital, and

�z; as mentioned above, is the average gross growth rate of ydt (these last two variables

are determined by the model, not by the government). The term mt is a random shock to

monetary policy that follows mt = �m"mt where "mt is distributed according to N (0; 1).The intuition behind the policy rules, motivated on a large body of empirical literature

(Orphanides, 2002), is that a good way to describe the behavior of the monetary authority is

to think about central banks as setting up the (short-term) interest rate as a function of the

past interest rate, Rt�1, the deviation of in�ation from a target, and the deviation of output

growth from the long-run average growth rate.

13

2.5. Aggregation

With some additional work, we can sum up the behavior of all agents in the economy to �nd

expressions for the remaining aggregate variables, including aggregate demand:

ydt = ct + xt + ��1t � [ut] kt�1

aggregate supply:

yst =At (utkt�1)

� �ldt �1�� � �zt

vpt

where:

vpt =

Z 1

0

�pitpt

��"di

and aggregate labor supply ldt = lt=vwt where

vwt =

Z 1

0

�wjtwt

���dj

The terms vpt and vwt represent the loss of e¢ ciency induced by price and wage dispersion.

Since all households and �rms are symmetrical, a social planner would like to set every price

and wage at the same level. However, the nominal rigidities prevent this socially optimal

solution. By the properties of the index under Calvo�s pricing, vpt and vwt evolve as:

vpt = �p

���t�1�t

��"vpt�1 + (1� �p)�

��"t

vwt = �w

�wt�1wt

��wt�1�t

���vwt�1 + (1� �w) (�

w�t )

�� :

2.6. Equilibrium

The equilibrium of this economy is characterized by the �rst order conditions of the household,

the �rst order conditions of the �rms, the policy rule of the government, and the aggregate

variables that we just presented, with a total of 21 equations (the web appendix has the list of

all of these 21 equations). Formally, we can think about them as a set of nonlinear functional

equations where the unknowns are the functions that determine the evolution over time of

the variables of the model. The variables that enter as arguments of these functions are the

19 state variables that determine the situation of the system at any point in time.

A simple inspection of the system reveals that it does not have an analytical solution. In-

stead, we need to resort to some type of numerical approximation to solve it. The literature on

how to do so is enormous, and we do not review it here (see, instead, the comprehensive text-

14

book by Judd, 1998). In previous work (Aruoba, Fernández-Villaverde, and Rubio-Ramírez,

2006), we convinced ourselves that a nice compromise between speed and accuracy in the

solution of systems of this sort can be achieved by the use of perturbation techniques.

The foundation of perturbation methods is to substitute the original system by a suitable

variation of it that allows for an analytically (or at least numerically) exact solution. In our

particular case, this variation of the problem is to set the standard deviations of all of the

model�s shocks to zero. Under this condition, the economy converges to a balanced growth

path, where all the variables grow at a constant (but potentially di¤erent) rate. Then, we

use this solution and a functional version of the implicit function theorem to compute the

leading coe¢ cients of the Taylor expansion of the exact solution.

Traditionally, the literature has focused on the �rst order approximation to the solution

because it generates a linear representation of the model whose likelihood is easily evaluated

using the Kalman �lter. Since this chapter is a brief introduction to the New Macroecono-

metrics, we will follow that convention. However, we could also easily compute a second order

approximation. Higher order approximations have the advantage of being more accurate and

of capturing precautionary behavior among agents induced by variances. We have found

in our own research that in many problems (but not in all!), those second order terms are

important.

Further elaborating this point, note that we end up working with an approximation of

the solution of the model and not with the solution itself raises an interesting technical issue.

Even if we were able to evaluate the likelihood implied by that solution, we would not be

evaluating the likelihood of the exact solution of the model but the likelihood implied by the

approximated solution of the model. Both objects may be quite di¤erent and some care is

required when we proceed to perform inference (Fernández-Villaverde, Rubio-Ramírez, and

Santos, 2006).

However, before digging into the details of the solution method, and since we have tech-

nological progress, we need to rescale nearly all variables to make them stationary. To do so,

de�ne ect = ctzt, e�t = �tzt, ert = rt�t, eqt = qt�t, ext = xt

zt, ewt = wt

zt, ew�t = w�t

zt, ekt = kt

zt�t, andeydt = ydt

zt. Also note that �c = �x = �w = �w� = �yd = �z. Furthermore, we normalize u = 1

in the steady state by setting �1 = er; eliminating it as a free parameter.For each variable vart, we de�nedvart = vart � var, as the deviation with respect to the

steady state. Then, the states of the model St are given by:

St =

0@ b�t�1; bewt�1; bg1t�1; bg2t�1;bekt�1; bRt�1;beydt�1;bect�1; bvpt�1; bvwt�1;beqt�1; bef t�1; bext�1; be�t�1;bezt�1; z�;t�1; bdt�1; b't�1; zA;t�11A0

;

15

and the exogenous shocks are "t = ("�;t; "d;t; "';t; "A;t; "m;t)0 :

From the output of the perturbation, we build the law of motion for the states:

St+1 = s1

�S0t; "

0t

�0(6)

where s1 is a 19�24 matrix, and the law of motion for the observables:

Yt = o1 (S 0t; "0t)0 (7)

where St =�S0t; S

0t�1; "

0t�1

�and o1 is a 5�48 matrix. We include lagged states in our

observation equation, because some of our data will appear in �rst di¤erences. Equivalently,

we could have added lagged observations to the states to accomplish the same objective. Also,

note that the function (7) includes a constant that captures the mean of the observables as

implied by the equilibrium of the model.

Our observables are the �rst di¤erences of the relative price of investment, output, real

wages, in�ation, and the federal funds rate, or in our notation:

Yt =�4 log ��1t ;4 log yt;4 logwt; log �t; logRt

�0:

While the law of motion for states is unique (or at least belonging to an equivalence class

of representations), the observation equation depends on what the econometrician actually

observes or chooses to observe. Since little is known about how those choices a¤ect our

estimation, we have selected the time series that we �nd particularly informative for our

purposes (see Guerrón-Quintana, 2008, for a detailed discussion).

For observables, we can only select a number of series less than or equal to the number of

shocks in the model. Otherwise, the model will be stochastically singular, i.e., we could write

the extra observables as deterministic functions of the other observables and the likelihood

would be -1. In the jargon of macroeconomics, these functions are part of the equilibriumcross-equation restrictions.

A popular way around the problem of stochastic singularity has been to assume that the

observables come with measurement error. This assumption delivers, by itself, one shock per

observable. There are three justi�cations for measurement error. First, statistical agencies

make mistakes. Measuring U.S. output or wages is a daunting task, undertaken with ex-

tremely limited resources by di¤erent government agencies. Despite their remarkable e¤orts,

statistical agencies can only provide us with an estimate of the series we need. Second, there

are di¤erences between the de�nitions of variables in the theory and in the data, some caused

by divergent methodological choices (see, for instance, the discussion in appendix B about

16

how to move from nominal output into real output), some caused by the limitations of the

data the statistical agency can collect. Finally, measurement errors may account for parts of

the dynamic of the data that are not captured by the model.

On the negative side, including measurement error complicates identi�cation and faces the

risk that the data ends up being explained by this measurement error and not by the model

itself. In our own research we have estimated DSGE models with and without measurement

error, according to the circumstances of the problem. In this chapter, since we have a rich

DSGE economy with many sources of uncertainty, we assume that our data come without

measurement error. This makes the exercise more transparent and easier to follow.

2.7. The Likelihood Function

Equations (6) and (7) plus the de�nition of St are the state space representation of a dynamic

model. It is well understood that we can use this state representation to �nd the likelihood

L (Y1:T ; ) of our DSGE model, where we have stacked all parameter values in:

=��; h; �; #; �; �; "; �; �; �w; �w; �p; �p;�2; R; y; �;�;��;�A; �d; �'; ��; �d; �A; �m; �'

:

and where Y1:T is the sequence of data from period 1 to T

To perform this evaluation of the likelihood function given some parameter values, we

start by factorizing the likelihood function as:

L (Y1:T ; ) =TYt=1

L (YtjY1:T�1; )

Then:

L (Y1:T ; ) =ZL (Y1jS0; ) dS0

TYt=2

ZL (YtjSt; ) p (StjY1:T�1; ) dSt (8)

If we know St, computing L (YtjSt; ) is conceptually simple since, conditional on St; themeasurement equation (7) is a change of variables from "t to Yt: Similarly, if we know S0,

it is easy to compute L (Y1jS0; ) with the help of (6) and (7): Thus, all that remains toevaluate the likelihood is to �nd the sequence fp (StjY1:t�1; )gTt=1 and the initial distributionp (S0; ).

This second task is comparatively simple. There are two procedures for doing so. First,

in the case where we can characterize the ergodic distribution, for example, in linearized

models, we can draw directly from it (this procedure takes advantage of the stationarity of

the model achieved by the rescaling of variables). In our model, we just need to set the states

to zero (remember, they are expressed in di¤erences with respect to the steady state) and

17

specify their initial variance-covariance matrix. Second, if we cannot characterize the ergodic

distribution, we can simulate the model for a large number of periods (250 for a model like

ours may su¢ ce). Under certain regularity conditions, the values of the states at period 250

are a draw from the ergodic distribution of states. Consequently, the only remaining barrier is

to characterize the sequence of conditional distributions fp (StjY1:T�1; )gTt=1 and to computethe integrals in (8).

Since a) we have performed a �rst order perturbation of the equilibrium equations of the

problem and generated a linear state space representation, and b) our shocks are normally

distributed, the sequence fp (StjY1:T�1; )gTt=1 is composed of conditionally Gaussian distri-butions. Hence, we can use the Kalman �lter, �nd fp (StjY1:T�1; )gTt=1 ; and evaluate thelikelihood in a quick and e¢ cient way. A version of the Kalman �lter applicable to our setup

is described in appendix B. The Kalman �lter is extremely e¢ cient: it takes less than one

second to run for each combination of parameter values.

However, in many cases of interest, we may want to perform a higher order perturbation

or the shocks to the model may not be normally distributed. Then, the components of

fp (StjY1:T�1; )gTt=1 would not be conditionally Gaussian. In fact, in general, it will notbelong to any known parametric family of density functions. In all of those cases, we can

track the desired sequence fp (StjY1:T�1; )gTt=1 using SMC methods (see Appendix A).Once we have the likelihood of the model L (Y1:T ; ) ; we combine it with a prior density

for the parameters p () to form a posterior

p (jY1:T ) / L (Y1:T ; ) p () :

Since the posterior is also di¢ cult to characterize (we do not even have an expression for the

likelihood, only an evaluation), we generate draws from it using a random walk Metropolis-

Hastings algorithm. The scaling matrix of the proposal density will be selected to generate

the appropriate acceptance ratio of proposals (Roberts, Gelman, and Gilks, 1997).

We can use the resulting empirical distribution to obtain point estimates, variances, etc.,

and to build the marginal likelihood that is a fundamental component in the Bayesian com-

parison of models (see Fernández-Villaverde and Rubio-Ramírez, 2004, for an application of

the comparison of dynamic models in economics). An important advantage of having draws

from the posterior is that we can also compute other objects of interest, like the posterior of

the welfare cost of business cycle �uctuations or the posterior of the responses of the economy

to an unanticipated shock. These objects are often of more interest to economists than the

parameter themselves.

18

3. Empirical Analysis

Now we are ready to take our model to the data and obtain point estimates and posterior

distributions. Again, because of space limitations, our analysis should be read more as an

example of the type of exercises that can be implemented than as an exhaustive investigation

of the empirical properties of the model. Furthermore, once we have parameter posteriors, we

can undertake many additional exercises of interest, like forecasting (where is the economy

going?), counterfactual analysis (what would have happen if?), policy design (what is the

best way to respond to shocks?), and welfare analysis (what is the cost of business cycle

�uctuations?).

We estimate our New Keynesian model using �ve time series for the U.S. economy: 1)

the relative price of investment with respect to the price of consumption, 2) real output per

capita growth, 3) real wages per capita, 4) the consumer price index, and 5) the federal funds

rate (the interest rate at which banks lend balances at the Federal Reserve System to each

other, usually overnight). Our sample goes from 1959Q1 - 2007Q1. Appendix B explains how

we construct these series.

Before proceeding further, we specify priors for the parameters. We will have two sets

of priors that will provide us with two sets of posteriors. First, we adopt �at priors for all

parameters except a few that we �x at some predetermined values. These �at priors are

modi�ed only by imposing boundary constraints to make the priors proper and to rule out

parameter values that are incompatible with the model (i.e., a negative value for a variance

or Calvo parameters outside the unit interval). Second, we use a set of priors nearly identical

to the one proposed by Smets and Wouters (2007) in a highly in�uential study in which they

estimate a similar DSGE model to ours.

Our �rst exercise with �at priors is motivated by the observation that, with this prior,

the posterior is proportional to the likelihood function.7 Consequently, our Bayesian results

can be interpreted as a classical exercise where the mode of the likelihood function (the point

estimate under an absolute value loss function for estimation) is the maximum likelihood

estimate. Moreover, a researcher who prefers alternative priors can always reweight the

draws from the posterior using importance sampling to encompass his favorite priors. We do

not argue that our �at priors are uninformative. After a reparameterization of the model,

a �at prior may become highly curved. Instead, we have found in our research that an

estimation with �at priors is a good �rst step to learn about the amount of information

carried by the likelihood and a natural benchmark against which to compare results obtained

7There is a small quali�er: the bounded support of some of the priors. We can eliminate this smalldi¤erence by thinking about those bounds as frontiers of admissible parameter values in a classical perspective.

19

with non�at priors. The main disadvantage of this �rst exercise is that, in the absence of

further information, it is extremely di¢ cult to get the McMc to converge and characterize

the posterior accurately. The number of local modes is so high that the chain wanders away

for the longest time without settling into the ergodic distribution.

In comparison, our second exercise, with Smets and Wouters�(2007) priors, is closer to

the exercises performed at central banks and other policy institutions. The priors bring a

considerable amount of additional information that allows us to achieve much more reasonable

estimates. However, the use of more aggressive priors requires a careful preparation by the

researcher, and in the case of institutional models, a candid conversation with the principal

regarding the assumptions about parameters she feels comfortable with.

3.1. Flat Priors

Now we present results from our exercise with �at priors. Before reporting results, we �x

some parameter values at the values reported in Table 1. In a Bayesian context, we can think

about �xing parameter values as having Dirac priors over them.

Table 1: Fixed Parameters

� h # � � " � � � �2

0:99 0:9 9 1:35 0:015 0:30 8 8 30 0 0:001

Macroeconomists �x some parameters for di¤erent reasons. One reason is related to

the amount of data: being too ambitious in estimating all the parameters in the model

with limited data sometimes delivers posteriors that are too spread for useful interpretation

and policy analysis. A related reason is that some parameters are poorly identi�ed with

macro data (for example, the elasticity of output with respect to capital), while, at the same

time, we have good sources of information from micro data not used in the estimation. It

seems, then, reasonable to set these parameters at the conventional values determined by

microeconometricians. In our �rst exercise with �at priors, the number of parameters that

we need to �x is relatively high, 11. Otherwise, the likelihood has too many local peaks, and

it turns out to be extremely di¢ cult to design a successful McMc.

We brie�y describe some of these parameter values in Table 1. The discount factor, �,

is �xed at 0.99, a number close to one, to match the low risk free rate observed in the U.S.

economy (the gross risk free rate depends on the inverse of � and the growth rate of the

economy). Habit persistence, h = 0:9; arises because we want to match the evidence of

a sluggish adjustment of consumption decisions to shocks. The parameter # pins down a

Frisch elasticity of 0.74, a value well within the range of the �ndings of microeconometrics

20

(Browning, Hansen, and Heckman, 1999). Our choice of �, the depreciation rate, matches

the capital-output ratio in the data. The elasticity of capital to output, � = 0:3, is informed

by the share of national income that goes to capital. The elasticity of substitution across

intermediate goods, ", accounts for the microeconomic evidence on the average mark-up of

U.S. �rms, which is estimated to be around 10-15 percent (the mark-up in our model is

approximately "=("� 1)). By symmetry, we pick the same value for �. We take a high valuefor the adjustment cost �, 30, to dampen the �uctuations of investment in the model and

get them closer to the behavior of investment in the data. We set � to zero, since we do not

have information on pure pro�ts by �rms. Luckily, since we do not have entry and exit of

�rms in the model, the parameter is nearly irrelevant for equilibrium dynamics. Finally, the

parameter controlling money demand � does not a¤ect the dynamics of the model because

the monetary authority will supply as much money as required to implement the nominal

interest rate determined by the policy rule.

We summarize the posterior distribution of the parameters in Table 2, where we report

the median of each parameter and its 5 and 95 percentile values and in Figure 1, where we

plot the histograms of the draws of each parameter. The results are based on a chain of 75,000

draws, initialized after a considerable amount of search for good initial starting values, and a

30 percent acceptance rate. We performed standard analysis of convergence of the chain to

ensure that the results are accurate.

Table 2: Median Estimated Parameters (5 and 95 percentile in parentheses)

�p � �w �w R y � �

0:72[0:68;0:77]

0:94[0:81;0:99]

0:62[0:56;0:70]

0:92[0:73;0:99]

0:87[0:83;0:90]

0:99[0:59;1:73]

2:65[2:10;3:67]

1:012[1:011;1:013]

�d �' �A �d �' �� �e �� �A

0:14[0:02;0:27]

0:91[0:86;0:93]

�4:45[�4:61;�4:29]

�2:53[�2:64;�2:43]

�2:29[�2:59;�1:74]

�5:49[�5:57;�5:40]

�5:97[�6:09;�5:84]

3e� 3[0:002;0:004]

1e� 4[0:00;0:0003]

[FIGURE 1 HERE]

We highlight a few results. The Calvo parameter for prices, �p; is closely estimated to be

around 0.72. This indicates that �rms revise their pricing decisions around 3.5 quarters on

average, a number that seems close to a natural benchmark of a yearly pricing cycle in many

�rms complemented with some �rms that revise their prices more often. The indexation level,

�, of 0.94, has a higher degree of uncertainty but also suggests that �rms respond quickly

to in�ation. High indexation generates a high level of in�ation inertia, and with it, a bigger

role for monetary policy. The Calvo parameter for wages, �w, implies wage decisions every

2.6 quarters, also with high indexation.

21

The parameters of the policy rule show that the monetary authority smooths to a large

degree the evolution of the federal funds rate, R is equal to 0.87, cares about the growth

level of the economy, y is equal to 0:99; and �nally, is rather aggressive against in�ation, �is equal to 2.65. This last parameter value is important, because it implies that the monetary

authority respects the �Taylor principle� that requires nominal interest rates to rise more

than in�ation rises. Otherwise, the real interest rate falls, in�ation rises, and we generate

bad policy outcomes due to the indeterminacy of equilibria (Lubick and Schorfheide, 2004).

The in�ation target of 1 percent per quarter is higher than the stated goals of the Federal

Reserve but consistent with the behavior of in�ation in the sample.

The standard deviations of the �ve shocks in the economy show the importance of the

preference shocks and the higher importance of the neutral technological shock in comparison

with the investment-speci�c technological shock in accounting for �uctuations in the economy.

In comparison, the posterior clearly indicated that the average growth of the economy is

driven by investment-speci�c technological change, whose mean, ��; is an order of magnitude

bigger than neutral technological change, �A. These �ndings are similar to the ones obtained

using a calibration approach by Greenwood, Herkowitz, and Krusell (1997).

3.2. Smets and Wouters�(2007) Priors

In our second estimation exercise, we incorporate priors for the parameters of the model based

on the work of Smets and Wouters (2007). First, as we did before, we �x some parameter

values. In this case, since the priors help to achieve identi�cation, we can free 6 out of

the 11 parameters �xed in the �rst exercise. The remaining 5 �xed parameters are those

still di¢ cult to identify with aggregate data and are summarized in Table 3. Note that,

to maintain comparability with Smets and Wouters� paper, we increase the elasticities of

substitution, " and �; to 10, an increase that has a minuscule e¤ect on the dynamics of the

data.

Table 3: Fixed Parameters

� " � � �2

0:025 10 10 0 0:001

We modify several of the priors of Smets and Wouters to induce better identi�cation and

to accommodate some of the di¤erences between their model and ours. Instead of a detailed

discussion of the reasons behind each prior, it is better to highlight that the priors are common

in the literature and are centered around values that are in the middle of the usual range

of estimates, using both macro and micro data, and diverse econometric methodologies. We

summarize our priors in Table 4.

22

Table 4: Priors

100���1 � 1

�h �p � �w

Ga(0:25; 0:1) Be(0:7; 0:1) N (9; 3) Be (0:5; 0:1) Be (0:5; 0:15) Be (0:5; 0:1)

�w R y � 100(�� 1) #

Be (0:5; 0:1) Be (0:75; 0:1) N (0:12; 0:05) N (1:5; 0:125) Ga (0:95; 0:1) N(1; 0:25)

� � �d �' exp(�A) exp(�d)

N (4; 1:5) N (0:3; 0:025) Be (0:5; 0:2) Be (0:5; 0:2) IG (0:1; 2) IG (0:1; 2)

exp(�') exp(��) exp(�e) 100�� 100�A

IG (0:1; 2) IG (0:1; 2) IG (0:1; 2) N (0:34; 0:1) N (0:178; 0:075)

We generate 75,000 draws from the posterior using our Metropolis-Hastings, also after an

exhaustive search for good initial parameters of the chain. In this case, the search is notably

easier than in the �rst exercise, showing the usefulness of the prior in achieving identi�cation

and a good behavior of the simulation. Table 5 reports the posterior medians, and the 5 and

95 percentile values of the 23 estimated parameters of the model, while Figure 2 plots the

histograms of each parameter.

Table 5: Median Estimated Parameters (5 and 95 percentile in parentheses)

� h # � �

0:998[0:997;0:999]

0:97[0:95;0:98]

8:92[4:09;13:84]

1:17[0:74;1:61]

9:51[7:47;11:39]

0:21[0:17;0:26]

�p � �w �w R y

0:82[0:78;0:87]

0:63[0:46;0:79]

0:68[0:62;0:73]

0:62[0:44;0:79]

0:77[0:74;0:81]

0:19[0:13;0:27]

� � �d �' �A �d

1:29[1:02;1:47]

1:010[1:008;1:011]

0:12[0:04;0:22]

0:93[0:89;0:96]

�3:97[�4:17;�3:78]

�1:51[�1:82;�1:11]

�' �� �e �� �A

�2:36[�2:76;�1:74]

�5:43[�5:52;�5:35]

�5:85[�5:94;�5:74]

3:4e� 3[0:003;0:004]

2:8e� 3[0:002;0:004]

[FIGURE 2 HERE]

The value of the discount factor, �, goes very close to 1. This result is typical in DSGE

models. The growth rate of the economy and the log utility function for consumption induce

a high risk free real interest rate even without any discounting. This result is di¢ cult to

reconcile with an observed low average real interest rate in the U.S. Hence, the likelihood

wants to push � as close as possible to 1 to avoid an even worse �t of the data. We �nd a

quite high level of habit, 0.97, which diverges from other �ndings in the literature that hover

around 0.7. The Frisch elasticity of labor supply of 0.85 (1/1.17) is similar to the one we

23

�xed in the �rst estimation exercise, a reassuring result since DSGE models have often been

criticized for relying on implausible high Frisch elasticities.

The adjustment cost of investment, �, is high, 9.51, although lower than the value we

�xed in the �rst exercise. Since our speci�cation of adjustment costs was parsimonious, this

result hints at the importance of further research on the nature of investment plans by �rms.

The parameter � is centered around 0.2. This result, which coincides with the �ndings of

Smets and Wouters, is lower than in other estimates, but it is hard to interpret because the

presence of monopolistic competition complicates the mapping between this parameter and

observed income shares in the national income and product accounts.

The Calvo parameter for price adjustment, �p; is 0.82 (an average �ve-quarter pricing

cycle) and the indexation level � is 0.63, although the posterior for this parameter is quite

spread. The Calvo parameter for wage adjustment, �w, is 0.68 (wage decisions are revised

every three quarters), while the indexation, �w; is 0.62. We see how the data push us

considerably away from the prior. There is information to be learned from the observations.

However, at the same time, the median of the posterior of the nominal rigidities parameters

is relatively di¤erent from the median in the case with �at priors. This shows how more

informative priors do have an impact on our inference, in particular in parameter values of

key importance for policy analysis as �p and �w. We do not necessarily judge this impact in

a negative way. If the prior is bringing useful information into the table (for example, from

micro data on individual �rms�price changes, as in Bils and Klenow, 2004), this is precisely

what we want. Nevertheless, the researcher and the �nal user of the model must be aware of

this fact.

The parameters of the policy rule� R; �; y;�

are standard, with the only signi�cant

di¤erence that now the Fed is less responsive to in�ation, with the coe¢ cient � = 1:29:

However, this value still respects the Taylor principle (even the value at the 5 percentile

does). The Fed smooths the interest rate over time ( R is estimated to be 0.79) and responds

actively to in�ation ( R is 1.25) and weakly to the output growth gap ( y is 0.19). We

estimate that the Fed has a target for quarterly in�ation of 1 percent.

The growth rates of the investment-speci�c technological change, ��, and of the neutral

technology, �A, are roughly equal. The estimated average growth rate of the economy in

per capita terms, (�A + ���) = (1� �) is 0.43 percent per quarter, or 1.7 percent annually,

roughly the historical mean in the U.S. in the period 1865-2007.

24

4. Lines of Further Research

While the previous pages o¤ered a snapshot of where the frontier of the research is, we want to

conclude by highlighting three avenues for further research. Our enumeration is only a small

sampler from the large menu of items to be explored in the New Macroeconometrics. We

could easily write a whole chapter just signaling areas of investigation where entrepreneurial

economists can spend decades and yet leave most of the territory uncharted.

First, we are particularly interested in the exploration of �exotic preferences� that go

beyond the standard expected utility function used in this chapter (Backus, Routledge, and

Zin, 2005). There is much evidence that expected utility functions have problems accounting

for many aspects of the behavior of economic agents, in particular those aspects determining

asset prices. For example, expected utility cannot explain the time inconsistencies that we

repeatedly see at the individual level (how long did you stick with your last diet?). Recently,

Bansal and Yaron (2004) and Hansen and Sargent (2007) have shown that many of the asset

pricing puzzles can be accounted for if we depart from expected utility. However, these authors

have used little formal econometrics and we do not know which of the di¤erent alternatives

to expected utility will explain the data better. Therefore, the estimation of DSGE models

with �exotic preferences� seems an area where the interaction between empirical work and

theory is particularly promising.

Second, we would like to relax some of the tight parametric restrictions of the model,

preferably within a Bayesian framework. DSGE models require many auxiliary parametric

assumptions that are not central to the theory. Unfortunately, many of these parametric as-

sumptions do not have a sound empirical foundation. But picking the wrong parametric form

may have horrible consequences for the estimation of dynamic models. These concerns mo-

tivate the study of how to estimate DSGE models combining parametric and nonparametric

components and, hence, conducting inference that is more robust to auxiliary assumptions.

Finally, a most exciting frontier is the integration of microeconomic heterogeneity within

estimated DSGE models. James Heckman and many other econometricians have shown be-

yond reasonable doubt that individual heterogeneity is the de�ning feature of micro data (see

Browning, Hansen, and Heckman, 1999). Our macro models need to move away from the

basic representative agent paradigm and include richer con�gurations. Of course, this raises

the di¢ cult challenge of how to e¤ectively estimate these economies, since the computation

of the equilibrium dynamics of the model is a challenge by itself. However, we are optimistic:

advances in computational power and methods have allowed us to do things that were un-

thinkable a decade ago. We do not see any strong reasons why estimating DSGE models with

heterogeneous agents will not be any di¤erent in a few years.

25

5. Appendix A: Broader Context and Background

5.1. Policy-Making Institutions and DSGE Models

We mentioned in the main text that numerous policy institutions are using estimated DSGE

models as an important input for their decisions. Without being exhaustive, we mention

the Federal Reserve Board (Erceg, Guerrieri, and Gust, 2006), the European Central Bank

(Christo¤el, Coenen, and Warne, 2007), the Bank of Canada (Murchison and Rennison,

2006), the Bank of England (Harrison et al., 2005), the Bank of Sweden (Adolfson et al.,

2005), the Bank of Finland (Kilponen and Ripatti, 2006 and Kortelainen, 2002), and the

Bank of Spain (Andrés, Burriel, and Estrada, 2006), among several others.

5.2. Sequential Monte Carlo Methods and DSGE Models

The solution of DSGE models can be written in terms of a state space representation. Often,

that state space representation is nonlinear and/or non-Gaussian. There are many possible

reasons for this. For example, we may want to capture issues, such as asymmetries, threshold

e¤ects, big shocks, or policy regime switching that are approximated very poorly (or not

at all) by a linear solution. Also, one of the key issues in macroeconomics is the time-

varying volatility in time series. McConnell and Pérez-Quirós (2000) and Kim and Nelson

(1998) presented extremely de�nitive evidence that the U.S. economy had been more stable

in the 1980s and 1990s than before. Fernández-Villaverde and Rubio-Ramírez (2007) and

Justiniano and Primiceri (2008) demonstrated that DSGE models, properly augmented with

non-Gaussian shocks, could account for this evidence.

Nonlinear and/or non-Gaussian state space representations complicate the �ltering prob-

lem. All the relevant conditional distributions of states became nonnormal and, except for

a few cases, we cannot resort to analytic methods to track them. Similarly, the curse of

dimensionality of numerical integration precludes the use of quadrature methods to compute

the relevant integrals of �ltering (those that appear in the Chapman-Kolmogorov formula

and in Bayes�theorem).

Fortunately, during the 1990s, a new set of tools was developed to handle this problem.

This set of tools has become to be known as sequential Monte Carlo (SMC) methods because

they use simulations period by period in the sample. The interested reader can see a general

introduction to SMC methods in Arulampalam et al. (2002), a collection of background ma-

terial and applications in Doucet, de Freitas, and Gordon (2001), and the �rst applications in

macroeconomics in Fernández-Villaverde and Rubio-Ramírez (2005 and 2007). The appendix

in that last paper also o¤ers references to alternative approaches.

The simplest SMCmethod is the Particle �lter, which can easily be applied to estimate our

26

DSGE model. This �lter replaces the conditional distribution of states fp (StjY1:T�1; )gTt=1by an empirical distribution ofN draws

�nsitjt�1

oNi=1

�Tt=1

from the sequence fp (StjY1:T�1; )gTt=1.These draws are generated by simulation. Then, by a trivial application of the law of large

numbers:

L (Y1:T ; ) '1

N

NXi=1

L�Y1jsi0j0;

� TYt=2

1

N

NXi=1

L�Ytjsitjt�1;

�where the subindex tracks the conditioning set (i.e., tjt � 1 means a draw at moment t

conditional on information until t� 1) and lower cases are realizations of a random variable.

The problem is then to draw from the conditional distributions fp (StjY1:T�1; )gTt=1.Rubin (1988) �rst noticed that such drawing can be done e¢ ciently by an application of

sequential sampling:

Proposition 1. Letnsitjt�1

oNi=1

be a draw from p (StjY1:T�1; ). Let the sequence fesitgNi=1be a draw with replacement from

nsitjt�1

oNi=1

where the resampling probability is given by

qit =L�Ytjsitjt�1;

�PN

i=1 L�Ytjsitjt�1;

� ;Then fesitgNi=1 is a draw from p (StjY1:T ; ).

Proposition 1 recursively uses a drawnsitjt�1

oNi=1

from p (StjY1:T�1; ) to drawnsitjt

oNi=1

from p (StjY1:T ; ). But this is nothing more than the update of our estimate of St to addthe information on yt that Bayes�theorem is asking for. Resampling ensures that this update

is done in an e¢ cient way.

Once we havensitjt

oNi=1, we draw N vectors of the 5 exogenous innovations in the DSGE

model (the two shocks to productivity, the two shocks to technology, and the monetary policy

shock) from the corresponding distributions and apply the law of motion for states to generatensit+1jt

oNi=1. This step, known as forecast, puts us back at the beginning of proposition 1, but

with the di¤erence that we have moved forward one period in our conditioning, implementing

in that way the Chapman-Kolmogorov equation. By going through the sample repeating these

steps, we complete the evaluation of the likelihood function. Künsch (2005) provides general

conditions for the consistency of this estimator of the likelihood function and for a central

limit theorem to apply.

The following pseudo-code summarizes the description of the algorithm:

27

Step 0, Initialization: Set t 1. Sample N valuesnsi0j0

oNi=1

from p (S0; ).

Step 1, Prediction: Sample N valuesnsitjt�1

oNi=1

usingnsit�1jt�1

oNi=1, the law of

motion for states and the distribution of shocks "t.

Step 2, Filtering: Assign to each draw�sitjt�1

�the weight !it in Proposition

1.

Step 3, Sampling: Sample N times with replacement fromnsitjt�1

oNi=1

using the

probabilities fqitgNi=1. Call each draw

�sitjt

�. If t < T set t t + 1 and go to

step 1. Otherwise stop.

6. Appendix B: Model and Computation

6.1. Computation of the Model

A feature of the New Macroeconometrics that some readers from outside economics may �nd

less familiar is that the researcher does not specify some functional forms to be directly esti-

mated. Instead, the economist postulates an environment with di¤erent agents, technology,

preferences, information, and shocks. Then, she concentrates on investigating the equilib-

rium dynamics of this environment and how to map the dynamics into observables. In other

words, economists are not satis�ed with describing behavior, they want to explain it from

�rst principles. Therefore, a necessary �rst step is to solve for the equilibrium of the model

given arbitrary parameter values. Once we have that equilibrium, we can build the associ-

ated estimating function (the likelihood, some moments, etc.) and apply data to perform

inference. Thus, not only is �nding the equilibrium of the model of key importance, but

doing it rapidly and accurately, since we may need to do it for many di¤erent combinations

of parameter values (for example, in an McMc simulation, we need to solve the model for

each draw of parameter values).

As we described in the main text, our solution algorithm for the model relies on the

perturbation of the equations that characterize the equilibrium of the model given some �xed

parameter values. The �rst step of the perturbation is to take partial derivatives of these

equations with respect to the states and control variables. This step, even if conceptually

simple, is rather cumbersome and requires an inordinate amount of algebra.

Our favorite approach for solving this step is to write a Mathematica program that gener-

ates the required analytic derivatives (which are in the range of several thousands) and that

writes automatic Fortran 95 code with the corresponding analytic expressions. This step is

28

crucial because, once we have paid the �xed cost of taking the analytic derivatives (which

takes several hours on a good desktop computer), solving the equilibrium dynamics for a new

combination of parameter values takes less than a second, since now we only need to solve

a numerical system in Fortran. The solution algorithm will be nested inside a Metropolis-

Hastings. For each proposed parameter value in the chain, we will solve the model, �nd the

equilibrium dynamics, use those dynamics to �nd the likelihood using the Kalman �lter, and

then accept or reject the proposal.

6.2. Kalman Filter

The implementation of the Kalman �lter to evaluate the likelihood function in a DSGE

model like ours follows closely that of Stengel (1994). We start by writing the �rst order

linear approximation to the solution of the model in a standard state space representation:

st = Ast�1 +B"t (9)

yt = Cst +D"t (10)

where st are the states of the model, yt are the observables, and "t � N (0; I) is the vector ofinnovations to the model.

We introduce some de�nitions. Let xtjt�1 = E (xtjYt�1) and xtjt = E (xtjYt) where Yt =fy1; y2; :::; ytg. Also, we have Pt�1jt�1 = E

�st�1 � st�1jt�1

� �st�1 � st�1jt�1

�0and Ptjt�1 =

E�st�1 � stjt�1

� �st�1 � stjt�1

�0:With this notation, the one-step-ahead forecast error is �t =

yt � Cxtjt�1.

We forecast the evolution of states:

stjt�1 = Axt�1jt�1 (11)

Since the possible presence of correlation in the innovations does not change the nature of

the �lter, it is still the case that

stjt = stjt�1 +K�t; (12)

where K is the Kalman gain at time t. De�ne variance of forecast as Vy = CPtjt�1C0 +DD0:

Then, the conditional likelihood is just:

logL = �n2log 2� � 1

2log det (Vy)�

1

2�tV

�1y �t

The last step is to update our estimates of the states. De�ne residuals �tjt�1 = st � stjt�1

29

and �tjt = st � stjt. Subtracting equation (11) from equation (9)

st � stjs�1 = A�st�1 � st�1jt�1

�+Bwt;

�tjt�1 = A�t�1jt�1 +Bwt (13)

Now subtract equation (12) from equation (9)

st � stjt = st � stjt�1 �K�Cxt +Dwt � Cxtjt�1

��tjt = �tjt�1 �K

�C�tjt�1 +Dwt

�: (14)

Note Ptjt�1 can be written as

Ptjt�1 = E�tjt�1�0tjt�1;

= E�A�t�1jt�1 +Bwt

� �A�t�1jt�1 +Bwt

�0= APt�1jt�1A

0 +BB0: (15)

For Ptjt we have:

Ptjt = E�tjt�0tjt= (I �KC)Ptjt�1 (I � C 0K 0) +KDD0K 0 �KDB0

�BD0K 0 +KCBD0K 0 +KDB0C 0K 0: (16)

The optimal gain minimizes Ptjt:

Kopt =�Ptjt�1C

0 +BD0� [V y + CBD0 +DB0C 0]�1

and, consequently, the updating equations are:

Ptjt = Ptjt�1 �Kopt

�DB0 + CPtjt�1

�;

stjt = stjt�1 +Kopt�t

and we close the iterations.

6.3. Construction of Data

When we estimate the model, we need to make the series provided by the national and income

product accounts (NIPA) consistent with the de�nition of variables in the theory. The main

adjustment that we undertake is to express both real output and real gross investment in

30

consumption units. Our DSGE model implies that there is a numeraire in terms of which all

the other prices need to be quoted. We pick consumption as the numeraire. The NIPA, in

comparison, uses an index of all prices to transform nominal GDP and investment into real

values. In the presence of changing relative prices, such as the ones we have seen in the U.S.

over the last several decades with the fall in the relative price of capital, NIPA�s procedure

biases the valuation of di¤erent series in real terms.

We map theory into data by computing our own series of real output and real investment.

To do so, we use the relative price of investment, de�ned as the ratio of an investment

de�ator and a de�ator for consumption. The denominator is easily derived from the de�ators

of nondurable goods and services reported in the NIPA. It is more complicated to obtain the

numerator because, historically, NIPA investment de�ators were poorly constructed. Instead,

we rely on the investment de�ator computed by Fisher (2006), a series that unfortunately ends

early in 2000Q4. Following Fisher�s methodology, we have extended the series to 2007Q1.

For the real output per capita series, we �rst de�ne nominal output as nominal consump-

tion plus nominal gross investment. We de�ne nominal consumption as the sum of personal

consumption expenditures on nondurable goods and services. We de�ne nominal gross invest-

ment as the sum of personal consumption expenditures on durable goods, private residential

investment, and nonresidential �xed investment. Per capita nominal output is equal to the

ratio between our nominal output series and the civilian noninstitutional population between

16 and 65. To obtain per capita values, we divide the previous series by the civilian noninsti-

tutional population between 16 and 65. Finally, real wages are de�ned as compensation per

hour in the nonfarm business sector divided by the CPI de�ator.

31

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34

0.6 0.7 0.80

5000

10000

15000

θp

0.7 0.8 0.90

5000

10000

15000

χ

0.6 0.7 0.80

5000

10000

15000

θw

0.4 0.6 0.80

0.5

1

1.5

2

2.5

x 104χw

0.8 0.85 0.90

5000

10000

γr

0.5 1 1.5 2 2.50

5000

10000

γy

2 3 40

5000

10000

γπ

1.01 1.012 1.0140

5000

10000

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0.2 0.40

2000

4000

6000

8000

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0.750.80.850.90.950

0.5

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10000

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5000

10000

σμ

-6.2 -6 -5.80

5000

10000

σe

2.5 3 3.5 4

x 10-3

0

5000

10000

Λμ

Figure 1: Posterior Distribution, Flat Priors

2 4 6

x 10-4

0

0.5

1

1.5

2

x 104Λa

0.996 0.9980

5000

10000

β

0.94 0.96 0.980

5000

10000

h

5 10 15 200

5000

10000

ψ

1 20

5000

10000

ϑ

0.15 0.2 0.250

5000

10000

α

5 100

5000

10000

15000

κ

0.75 0.8 0.850

5000

10000

θp

0.4 0.6 0.80

5000

10000

χ

0.60.650.70.750

5000

10000

θw

0.4 0.6 0.80

5000

10000

χw

0.7 0.80

5000

10000

γr

0.1 0.2 0.30

5000

10000

15000

γy

1.2 1.4 1.60

5000

10000

γπ

1.0081.011.0120

5000

10000

Π

0.2 0.40

5000

10000

15000

ρd

0.85 0.9 0.950

5000

10000

15000

ρφ

-4.4-4.2 -4 -3.8-3.60

5000

10000

σa

-2 -1.5 -10

5000

10000

15000

σd

-3 -2 -10

5000

10000

15000

σφ

-5.6 -5.40

5000

10000

15000

σμ

-6 -5.8 -5.60

5000

10000

15000

σe

2.5 3 3.5 4 4.5

x 10-3

0

5000

10000

Λμ

Figure 2: Posterior Distribution, Smets-Wouters Priors

2 4

x 10-3

0

5000

10000

15000

Λa